In childhood, people are often taught the fundamentals of counting by using their fingers. Counting from one to ten is one of many milestones a child achieves on their way to becoming educated members of society. We will review these basic facts on our way to gaining an understanding of alternate number systems.

The child is taught that the fingers and thumbs can be used to count from one to ten. Extending one finger represents a count of one; two fingers represents a count of two, and so on up to a maximum count of ten. No fingers (or thumbs) refers to a count of zero.

The child is later taught that there are certain symbols called digits that can be used to represent these counts. These digits are, of course:

0,1,2,3,4,5,6,7,8,9{\displaystyle 0,1,2,3,4,5,6,7,8,9\,\!}

Ten, of course, is a special case, since it is comprised of two digits.

The radix, or base of a counting system is defined as the number of unique digits in a given number system.

Back to our elementary example. We know that our hypothetical child can count from zero to ten using their fingers and thumbs. There are ten unique digits in this counting system, therefore the radix of our elementary counting system is ten.

We represent the radix of our counting system by putting the radix in subscript to the right of the digits. For example,

310{\displaystyle 3_{10}\,\!}

represents 3 in decimal (base 10).

Our special case (ten) illustrates a fundamental rule of our number system that was not readily apparent - what happens when the count exceeds the highest digit? Obviously, a new digit is added, to the left of our original digit which is "worth more", or has a higher weight than our original digit. (In reality, there *always* are digits to the left; we simply choose not to write those digits to the left of the first nonzero.)

It is widely believed that the decimal system that we find so natural to use is a direct consequence of a human being's ten fingers and thumbs being used for counting purposes. One could easily imagine that a race of intelligent, six-fingered beings could quite possibly have developed a base-six counting system. From this perspective, consider the hypothesis: the most intuitive number system for an entity is that for which some natural means of counting exists.

Since our focus is electronic and computer systems, we must narrow our focus from the human hand to the switch, arguably the most fundamental structure that can be used to represent a count.

The switch can represent one of two states; either open, or closed. If we return to our original definition of a digit, how many digits are required to represent the possible states of our switch? Clearly, the answer is 2. We use the binary digits zero and one to represent the open and closed states of the switch.

Counting in binary like base 10 can be accomplished with your fingers with some differences. In decimal base 10 each finger represents a number in the first digit 1, 2, 3, 4, 5, 6, 7, 8, 9 ,10 . When counting in binary each finger represents a separate digit so counting is more like this.

each number is a finger dont use thumbs for binary

value

decimal

binary

right left hand hand

4fingers

0

00000 00000

0000

1

00000 01000

0001

2

00000 01100

0010

3

00000 01110

0011

4

00000 01111

0100

5

00000 11111

0101

6

00010 11111

0110

7

00110 11111

0111

8

01110 11111

1000

9

11110 11111

1001

10

11111 11111

1010

WHOA, hold on, that pattern doesn't make sense? Well when you consider that each digit in binary is worth 2 times more than the previous (right to left always) then it makes sense. Lets compare number lines.

To convert from decimal to binary, you can repeatedly divide the decimal by two until the result of the division is zero. Starting from the rightmost bit, write 1 if the division has a remainder, zero if it does not. For example, to convert the decimal 74 into binary:

Each octal digit is representable by exactly three bits. This becomes obvious when you consider that the highest octal digit is seven, which can be represented in binary by 1112{\displaystyle 111_{2}\,\!}.

To convert a binary number to octal, group the bits in groups of three starting from the rightmost bit and convert each triplet to its octal equivalent.

The most commonly-used number system in computer systems is the hexadecimal, or more simply hex, system. It has a radix of 16, and uses the numbers zero through nine, as well as A through F as its digits:

Each hex digit is representable by exactly four bits. This becomes obvious when you consider that the highest hex digit represents fifteen, which can be represented in binary by 11112{\displaystyle 1111_{2}\,\!}.

To convert a binary number to hex, group the bits in groups of four starting from the rightmost bit and convert each group to its hex equivalent.